Molecules and Their Molecular Orbitals

Chapter 3. Diatomic Molecules and their Molecular Orbitals... 

We shall illustrate these rules first with H2 and then with other diatomic molecules. The same principles apply to polyatomic molecules, but their molecular orbitals are more complicated and their energies are harder to predict. Mathematical software for calculating the molecular orbitals and their energies is now widely available, and we shall show some of the results that it provides. 

The boranes are electron-deficient compounds (Section 3.8) we cannot write valid Lewis structures for them, because too few electrons are available. For instance, there are 8 atoms in diborane, so we need at least 7 bonds however, there are only 12 valence electrons, and so we can form at most 6 electron-pair bonds. In molecular orbital theory, these electron pairs are regarded as delocalized over the entire molecule, and their bonding power is shared by several atoms. In diborane, for instance, a single electron pair is delocalized over a B—H—B unit. It binds all three atoms together with bond order of 4 for each of the B—H bridging bonds. The molecule has two such bridging three-center bonds (9). 

As can be seen from the energy level structure diagram, the relative position of the HOMO and LUMO levels are not less important than the energy gap between them, since they control the possibility of charge injection. At this point, however, note, that a MO scheme is often used for illustration, but more properly the total energy states of the molecules and their radical cations and anions that may be subjected to electronic rearrangement have to be considered. Bearing this in mind, the measured values of redox potentials can be translated into the molecular orbital picture. 

Just as in the non-linear polyatomic-molecule case, the atomic orbitals which constitute a given molecular orbital must have the same symmetry as that of the molecular orbital. This means that o,%, and 8 molecular orbitals are formed, via LCAO-MO, from m=0, m= 1, and m= 2 atomic orbitals, respectively. In the diatomic N2 molecule, for example, the core orbitals are of o symmetry as are the molecular orbitals formed from the 2s and 2pz atomic orbitals (or their hybrids) on each Nitrogen atom. The molecular orbitals fonned from the atomic 2p i =(2px- i 2py) and the 2p+j =(2px + i 2py ) orbitals are of Jt symmetry and have m = -1 and +1. 

This success of density functional theory allows the whole question of bonding and structure to be formulated within an effective one-electron framework that is so beloved by chemists in their molecular orbital description of molecules and by physicists in their band theory description of solids. In this book I have tried to follow Einstein s dictum by simplifying the one-electron problem to the barest... 

Nowadays a wide variety of quantum-chemical programs are disposable, which permit to calculate with high accuracy the equilibrium geometry of the molecules and their energy of formation. Theoretical methods have been developed for analytical calculation of the first and second derivatives of energy [8,9], so that the force-constant matrix FHT and the harmonic frequencies can be extracted from the quantum-mechanical calculations. Since as a rule the molecular orbitals (MO) obtained by the quantum-mechanical methods are spread around the entire molecule, the corresponding quantum-mechanical force fields incorporate the important effects of the off-diagonal interactions. 

Although both atomic orbitals and molecular orbitals are one-electron wave functions, the shape and symmetry of the molecular orbitals are different from those of the atomic orbitals of the isolated atom. The molecular orbitals extend over the entire molecule, and their spatial symmetry must conform to that of the molecular framework. Of course, the electron distribution is not uniform throughout the molecular orbital. In depicting these orbitals, usually only the portions with substantial electron density are emphasized. 

We now have four molecular orbitals, o, cr2, crx and cry, one lowered in energy and one raised relative to the energy of the orbitals of the pair of hydrogen molecules. If we have four electrons in the system, the net result is repulsion. Thus two H2 molecules do not combine to form an H4 molecule. This is true whatever geometry we use in the combination. It shows us why molecules exist—when two molecules approach each other, the interaction of their molecular orbitals usually leads to repulsion. 

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